Effect of artificial periodicity in simulations of biomolecules under Ewald boundary conditions: a continuum electrostatics study

Multistate Method to Efficiently Account for Tautomerism and Protonation in Alchemical Free-Energy Calculations

The majority of drug-like molecules contain at least one ionizable group, and many common drug scaffolds are subject to tautomeric equilibria. Thus, these compounds are found in a mixture of protonation and/or tautomeric states at physiological pH. Intrinsically, standard classical molecular dynamics (MD) simulations cannot describe such equilibria between states, which negatively impacts the prediction of key molecular properties in silico. Following the formalism described by de Oliveira and co-workers (J. Chem. Theory Comput. 2019, 15, 424-435) to consider the influence of all states on the binding process based on alchemical free-energy calculations, we demonstrate in this work that the multistate method replica-exchange enveloping distribution sampling (RE-EDS) is well suited to describe molecules with multiple protonation and/or tautomeric states in a single simulation. We apply our methodology to a series of eight inhibitors of factor Xa with two protonation states and a series of eight inhibitors of glycogen synthase kinase 3β (GSK3β) with two tautomeric states. In particular, we show that given a sufficient phase-space overlap between the states, RE-EDS is computationally more efficient than standard pairwise free-energy methods.

Release of nanodiscs from charged nano-droplets in the electrospray ionization revealed by molecular dynamics simulations

Electrospray ionization (ESI) is essential for application of mass spectrometry in biological systems, as it prevents the analyte being split into fragments. However, due to lack of a clear understanding of the mechanism of ESI, the interpretation of mass spectra is often ambiguous. This is a particular challenge for complex biological systems. Here, we focus on systems that include nanodiscs as membrane environment, which are essential for membrane proteins. We performed microsecond atomistic molecular dynamics simulations to study the release of nanodiscs from highly charged nano-droplets into the gas phase, the late stage of ESI. We observed two distinct major scenarios, highlighting the diversity of morphologies of gaseous product ions. Our simulations are in reasonable agreement with experimental results. Our work provides a detailed atomistic view of the ESI process of a heterogeneous system (lipid nanodisc), which may give insights into the interpretation of mass spectra of all lipid-protein systems.

Review of Electrostatic Force Calculation Methods and Their Acceleration in Molecular Dynamics Packages Using Graphics Processors

Molecular dynamics (MD) simulations probe the conformational repertoire of macromolecular systems using Newtonian dynamic equations. The time scales of MD simulations allow the exploration of biologically relevant phenomena and can elucidate spatial and temporal properties of the building blocks of life, such as deoxyribonucleic acid (DNA) and protein, across microsecond (μs) time scales using femtosecond (fs) time steps. A principal bottleneck toward extending MD calculations to larger time scales is the long-range electrostatic force measuring component of the naive nonbonded force computation algorithm, which scales with a complexity of (N, number of atoms). In this review, we present various methods to determine electrostatic interactions in often-used open-source MD packages as well as the implementation details that facilitate acceleration of the electrostatic interaction calculation.

Computational Tools for Accurate Binding Free-Energy Prediction

A quantitative thermodynamic understanding of the noncovalent association of (bio)molecules is of central importance in molecular life sciences. An important quantity characterizing (bio)molecular association is the binding affinity or absolute binding free energy. In recent years, the computational prediction of absolute binding free energies has evolved considerably in terms of accuracy, computational speed, and user-friendliness. In this chapter, we first give an overview of how absolute free energies are defined and how they can be determined with computational means. We proceed with an outline of the theoretical basis of the two most reliable methods, potential of mean force, and double decoupling calculations. In particular, we describe how the sampling problem can be alleviated by application of restraints. Finally, we provide step-by-step instructions of how to set up corresponding molecular simulations with a commonly employed molecular dynamics simulation engine.

Recent Developments in Free Energy Calculations for Drug Discovery

The grand challenge in structure-based drug design is achieving accurate prediction of binding free energies. Molecular dynamics (MD) simulations enable modeling of conformational changes critical to the binding process, leading to calculation of thermodynamic quantities involved in estimation of binding affinities. With recent advancements in computing capability and predictive accuracy, MD based virtual screening has progressed from the domain of theoretical attempts to real application in drug development. Approaches including the Molecular Mechanics Poisson Boltzmann Surface Area (MM-PBSA), Linear Interaction Energy (LIE), and alchemical methods have been broadly applied to model molecular recognition for drug discovery and lead optimization. Here we review the varied methodology of these approaches, developments enhancing simulation efficiency and reliability, remaining challenges hindering predictive performance, and applications to problems in the fields of medicine and biochemistry.

On the importance of statistics in molecular simulations for thermodynamics, kinetics and simulation box size

Computational simulations, akin to wetlab experimentation, are subject to statistical fluctuations. Assessing the magnitude of these fluctuations, that is, assigning uncertainties to the computed results, is of critical importance to drawing statistically reliable conclusions. Here, we use a simulation box size as an independent variable, to demonstrate how crucial it is to gather sufficient amounts of data before drawing any conclusions about the potential thermodynamic and kinetic effects. In various systems, ranging from solvation free energies to protein conformational transition rates, we showcase how the proposed simulation box size effect disappears with increased sampling. This indicates that, if at all, the simulation box size only minimally affects both the thermodynamics and kinetics of the type of biomolecular systems presented in this work.

Generalized Moment Correction for Long-Ranged Electrostatics

Describing long-ranged electrostatics using short-ranged pair potentials is appealing because the computational complexity scales linearly with the number of particles. The foundation of the approach presented here is to mimic the long-ranged medium response by cancelling electric multipoles within a small cutoff sphere. We propose a rigorous and formally exact new method that cancels up to infinitely many multipole moments and is free of operational damping parameters often required in existing theories. Using molecular dynamics simulations of water with and without added salt, we discuss radial distribution functions, Kirkwood–Buff integrals, dielectrics, diffusion coefficients, and angular correlations in relation to existing electrostatic models. We find that the proposed method is an efficient and accurate alternative for handling long-ranged electrostatics as compared to Ewald summation schemes. The methodology and proposed parameterization are applicable also for dipole–dipole interactions.

String Method for Protein-Protein Binding Free-Energy Calculations

A powerful computational strategy to determine the equilibrium association constant of two macromolecules with explicit-solvent molecular dynamics (MD) simulations is the "geometric route", which considers the reversible physical separation of the bound complex in solution. Nonetheless, multiple challenges remain to render this type of methodology reliable and computationally efficient in practice. In particular, in one, formulation of the geometric route relies on the potential of mean force (PMF) for physically separating the two binding partners restrained along a straight axis, which must be selected prior to the calculation. However, practical applications indicate that the calculation of the separation PMF along the predefined rectilinear pathway may be suboptimal and slowly convergent. Recognizing that a rectilinear straight separation pathway is generally not representative of how the protein complex physically separates in solution, we put forth a novel theoretical framework for binding free-energy calculations, leaning on the optimal curvilinear minimum free-energy path (MFEP) determined from the string method. The proposed formalism is validated by comparing the results obtained using both rectilinear and curvilinear pathways for a prototypical host-guest complex formed by cucurbit[7]uril (CB[7]) binding benzene, and for the barnase-barstar protein complex. On the basis of multi-microsecond MD calculations, we find that the calculations following the traditional rectilinear pathway and the string-based curvilinear pathway agree quantitatively, but convergence is faster with the latter.

A fast and accurate computational method for the linear-combination-based isotropic periodic sum

An isotropic periodic sum (IPS) is a powerful technique to reasonably calculate intermolecular interactions for wide range of molecular systems under periodic boundary conditions. A linear-combination-based IPS (LIPS) has been developed to attain computational accuracy close to an exact lattice sum, such as the Ewald sum. The algorithm of the original LIPS method has a high computational cost because it needs long-range interaction calculations in real space. This becomes a performance bottleneck for long-time molecular simulations. In this work, the combination of an LIPS and fast Fourier transform (FFT) was developed, and evaluated on homogeneous and heterogeneous molecular systems. This combinational approach of LIPS/FFT attained computational efficiency close to that of a smooth particle mesh Ewald while maintaining the same high accuracy as the original LIPS. We concluded that LIPS/FFT has great potential to extend the capability of IPS techniques for the fast and accurate computation of many types of molecular systems.

Classical Drude Polarizable Force Field Model for Methyl Phosphate and Its Interactions with Mg2

Phosphate groups are essential components of nucleic acids and proteins, whose interactions with solvent, metal ions, and ionic side chains help control folding and binding. Methyl phosphate (MP) represents a simple analog of phosphate moieties that are post-translation modifications in proteins and present at the termini of nucleic acids, among other environments. In the present study, we optimized parameters for use in polarizable molecular dynamics simulations of MP in its mono- and dianionic forms, MP- ≡ CH3HPO4- and MP2- ≡ CH3PO42-, along with P i2- ≡ HPO42-, in the context of the classical Drude oscillator model. Parameter optimization was done in a manner consistent with the remainder of the Drude molecular mechanics force field, choosing atomic charges and polarizabilities to reproduce molecular properties from quantum mechanics as well as experimental hydration free energies. Optimized parameters were similar to existing dimethyl phosphate parameters, with a few significant differences. The developed parameters were then used to compute magnesium binding affinities in aqueous solution, using alchemical molecular dynamics free energy simulations. Good agreement with experiment was obtained, and outer sphere binding was shown to be predominant for MP- and MP2-.

Anisotropic reaction field correction for long-range electrostatic interactions in molecular dynamics simulations

Reaction-field (RF) methods have been extensively used in molecular dynamics simulations to efficiently compute long-range electrostatic interactions. They assume a continuous dielectric medium outside a certain cutoff, which has shown to be a reasonable approximation in many cases. However, lattice sum or fast multipole methods are nowadays often used instead, which treat long-range interactions explicitly but may introduce different artefacts. In the following work, the major issue of RFs is addressed, i.e., their inability to account for inhomogeneity even in heterogenous environments (e.g., membranes or protein binding pockets). By using a first-order Laplace series expansion of the dielectric permittivity on the cutoff sphere, local anisotropic effects can be described in a simple form. It is shown that the resulting boundary-value problem cannot be solved analytically, but instead a well-behaved approximative anisotropic reaction field (ARF) is introduced, which preserves coordinate invariance and approaches the standard RF solution for homogeneous systems. The comparison of RF to the state-of-the-art particle-particle particle-mesh (P₃M) method shows a difference in the orientation of molecules close to the interface between two different dielectrics (water/chloroform). It was found that the ARF leads to a correction of the orientational distribution toward the P₃M reference for planar and spherical interfaces.

Alchemical free energy calculations and umbrella sampling with local molecular field theory

Understanding the thermodynamic driving forces underlying any chemical process requires a description of the underlying free energy surface. However, computation of free energies is difficult, often requiring advanced sampling techniques. Moreover, these computations can be further complicated by the evaluation of any long-ranged interactions in the system of interest, such as Coulomb interactions in charged and polar media. Local molecular field theory is a promising approach to avoid many of the conceptual and computational difficulties associated with long-ranged interactions. We present frameworks for performing alchemical free energy calculations and non-Boltzmann sampling with local molecular field theory. We demonstrate that local molecular field theory can be used to perform these free energy calculations with accuracy comparable to traditional methodologies while eliminating the need for explicit treatment of long-ranged interactions in simulations.

Assigning crystallographic electron densities with free energy calculations-The case of the fluoride channel Fluc

Approximately 90% of the structures in the Protein Data Bank (PDB) were obtained by X-ray crystallography or electron microscopy. Whereas the overall quality of structure is considered high, thanks to a wide range of tools for structure validation, uncertainties may arise from density maps of small molecules, such as organic ligands, ions or water, which are non-covalently bound to the biomolecules. Even with some experience and chemical intuition, the assignment of such disconnected electron densities is often far from obvious. In this study, we suggest the use of molecular dynamics (MD) simulations and free energy calculations, which are well-established computational methods, to aid in the assignment of ambiguous disconnected electron densities. Specifically, estimates of (i) relative binding affinities, for instance between an ion and water, (ii) absolute binding free energies, i.e., free energies for transferring a solute from bulk solvent to a binding site, and (iii) stability assessments during equilibrium simulations may reveal the most plausible assignments. We illustrate this strategy using the crystal structure of the fluoride specific channel (Fluc), which contains five disconnected electron densities previously interpreted as four fluoride and one sodium ion. The simulations support the assignment of the sodium ion. In contrast, calculations of relative and absolute binding free energies as well as stability assessments during free MD simulations suggest that four of the densities represent water molecules instead of fluoride. The assignment of water is compatible with the loss of these densities in the non-conductive F82I/F85I mutant of Fluc. We critically discuss the role of the ion force fields for the calculations presented here. Overall, these findings indicate that MD simulations and free energy calculations are helpful tools for modeling water and ions into crystallographic density maps.

Role of Counterions in Constant-pH Molecular Dynamics Simulations of PAMAM Dendrimers

Electrostatic interactions play a pivotal role in the structure and mechanism of action of most biomolecules. There are several conceptually different methods to deal with electrostatics in molecular dynamics simulations. Ionic strength effects are usually introduced using such methodologies and can have a significant impact on the quality of the final conformation space obtained. We have previously shown that full system neutralization can lead to wrong lipidic phases in the 25% PA/PC bilayer (J. Chem. Theory Comput. 2014,10, 5483-5492). In this work, we investigate how two limit approaches to the ionic strength treatment (implicitly with GRF or using full system neutralization with either GRF or PME) can influence the conformational space of the second-generation PAMAM dendrimer. Constant-pH MD simulations were used to map PAMAM's conformational space at its full pH range (from 2.5 to 12.5). Our simulations clearly captured the coupling between protonation and conformation in PAMAM. Interestingly, the dendrimer conformational distribution was almost independent of the ionic strength treatment methods, which is in contrast to what we have observed in charged lipid bilayers. Overall, our results confirm that both GRF with implicit ionic strength and a fully neutralized system with PME are valid approaches to model charged globular systems, using the GROMOS 54A7 force field.

Absolute Alchemical Free Energy Calculations for Ligand Binding: A Beginner's Guide

Many thermodynamic quantities can be extracted from computer simulations that generate an ensemble of microstates according to the principles of statistical mechanics. Among these quantities is the free energy of binding of a small molecule to a macromolecule, such as a protein. Here, we present an introductory overview of a protocol that allows for the estimation of ligand binding free energies via molecular dynamics simulations. While we focus on the binding of organic molecules to proteins, the approach is in principle transferable to any pair of molecules.

Explicit Solvent Hydration Benchmark for Proteins with Application to the PBSA Method

Explicit and implicit solvent models have a proven record of delivering hydration free energies of small, druglike solutes in reasonable agreement with experiment. Hydration of macromolecules, such as proteins, is to a large extent uncharted territory, with few results shedding light on quantitative consistency between different solvent models, let alone their ability to reproduce real water. In this work, based on extensive explicit solvent simulations employing TIP3P and SPC/E water models we analyze hydration free energy changes between fixed conformations of 5 diverse proteins, including large multidomain structures. For the two solvent models we find better agreement in electrostatic rather than nonpolar contributions (RMSE of 2.3 and 2.7 kcal/mol, respectively), even though absolute values of the latter are typically an order of magnitude smaller. We also highlight the importance of finite size corrections to relative protein hydration free energies, which turn out to be rather large, on the order of several kcal/mol, and are necessary for proper interpretation of results obtained under periodic boundary conditions. We further compare gathered data with predictions of the implicit solvent approach based on the Poisson equation and the surface or volume based nonpolar term. We find definitely lesser consistency than between the two explicit models (RMSE between implicit and TIP3 results of 11.3 and 8.4 kcal/mol for electrostatic and nonpolar contributions, respectively). In the process we determine the value of the protein dielectric constant and the geometric model for the dielectric boundary that provide for the best agreement. Finally, we evaluate the usefulness of surface and volume based models of nonpolar contributions to hydration free energy of large biomolecules.

All-Atom Continuous Constant pH Molecular Dynamics With Particle Mesh Ewald and Titratable Water

Development of a pH stat to properly control solution pH in biomolecular simulations has been a long-standing goal in the community. Toward this goal recent years have witnessed the emergence of the so-called constant pH molecular dynamics methods. However, the accuracy and generality of these methods have been hampered by the use of implicit-solvent models or truncation-based electrostatic schemes. Here we report the implementation of the particle mesh Ewald (PME) scheme into the all-atom continuous constant pH molecular dynamics (CpHMD) method, enabling CpHMD to be performed with a standard MD engine at a fractional added computational cost. We demonstrate the performance using pH replica-exchange CpHMD simulations with titratable water for a stringent test set of proteins, HP36, BBL, HEWL, and SNase. With the sampling time of 10 ns per replica, most pK_a's are converged, yielding the average absolute and root-mean-square deviations of 0.61 and 0.77, respectively, from experiment. Linear regression of the calculated vs experimental pK_a shifts gives a correlation coefficient of 0.79, a slope of 1, and an intercept near 0. Analysis reveals inadequate sampling of structure relaxation accompanying a protonation-state switch as a major source of the remaining errors, which are reduced as simulation prolongs. These data suggest PME-based CpHMD can be used as a general tool for pH-controlled simulations of macromolecular systems in various environments, enabling atomic insights into pH-dependent phenomena involving not only soluble proteins but also transmembrane proteins, nucleic acids, surfactants, and polysaccharides.

Strings-to-Rings Transition and Antiparallel Dipole Alignment in Two-Dimensional Methanols

Structural order emerging in the liquid state necessitates a critical degree of anisotropy of the molecules. For example, liquid crystals and Langmuir monolayers require rod- or disc-shaped and long-chain amphiphilic molecules, respectively, to break the isotropic symmetry of liquids. In this Letter we present results from molecular dynamics simulations demonstrating that in two-dimensional liquids, a significantly smaller degree of anisotropy is sufficient to allow structural organization. In fact, the condensed phase of the smallest amphiphilic molecule, methanol, confined between two, or adsorbed on, graphene sheets forms a monolayer characterized by long chains of molecules. Intrachain interactions are dominated by hydrogen bonds, whereas interchain interactions are dispersive. Upon a decrease in density toward a gaslike state, these strings are transformed into rings. The two-dimensional liquid phase of methanol undergoes another transition upon cooling; in this case, the order-disorder transition is characterized by a low-temperature phase in which the hydrogen bond dipoles of neighboring strings adopt an antiparallel orientation.

Long-ranged contributions to solvation free energies from theory and short-ranged models

Long-standing problems associated with long-ranged electrostatic interactions have plagued theory and simulation alike. Traditional lattice sum (Ewald-like) treatments of Coulomb interactions add significant overhead to computer simulations and can produce artifacts from spurious interactions between simulation cell images. These subtle issues become particularly apparent when estimating thermodynamic quantities, such as free energies of solvation in charged and polar systems, to which long-ranged Coulomb interactions typically make a large contribution. In this paper, we develop a framework for determining very accurate solvation free energies of systems with long-ranged interactions from models that interact with purely short-ranged potentials. Our approach is generally applicable and can be combined with existing computational and theoretical techniques for estimating solvation thermodynamics. We demonstrate the utility of our approach by examining the hydration thermodynamics of hydrophobic and ionic solutes and the solvation of a large, highly charged colloid that exhibits overcharging, a complex nonlinear electrostatic phenomenon whereby counterions from the solvent effectively overscreen and locally invert the integrated charge of the solvated object.

The good, the bad and the user in soft matter simulations

Molecular dynamics (MD) simulations have become popular in materials science, biochemistry, biophysics and several other fields. Improvements in computational resources, in quality of force field parameters and algorithms have yielded significant improvements in performance and reliability. On the other hand, no method of research is error free. In this review, we discuss a few examples of errors and artifacts due to various sources and discuss how to avoid them. Besides bringing attention to artifacts and proper practices in simulations, we also aim to provide the reader with a starting point to explore these issues further. In particular, we hope that the discussion encourages researchers to check software, parameters, protocols and, most importantly, their own practices in order to minimize the possibility of errors. The focus here is on practical issues. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Hydration Free Energies of Molecular Ions from Theory and Simulation

We present a theoretical/computational framework for accurate calculation of hydration free energies of ionized molecular species. The method is based on a molecular theory, 3D-RISM, combined with a recently developed pressure correction (PC+). The 3D-RISM/PC+ model can provide ∼3 kcal/mol hydration free energy accuracy for a large variety of ionic compounds, provided that the Galvani potential of water is taken into account. The results are compared with direct atomistic simulations. Several methodological aspects of hydration free energy calculations for charged species are discussed.

Scaling of Multimillion-Atom Biological Molecular Dynamics Simulation on a Petascale Supercomputer

A strategy is described for a fast all-atom molecular dynamics simulation of multimillion-atom biological systems on massively parallel supercomputers. The strategy is developed using benchmark systems of particular interest to bioenergy research, comprising models of cellulose and lignocellulosic biomass in an aqueous solution. The approach involves using the reaction field (RF) method for the computation of long-range electrostatic interactions, which permits efficient scaling on many thousands of cores. Although the range of applicability of the RF method for biomolecular systems remains to be demonstrated, for the benchmark systems the use of the RF produces molecular dipole moments, Kirkwood G factors, other structural properties, and mean-square fluctuations in excellent agreement with those obtained with the commonly used Particle Mesh Ewald method. With RF, three million- and five million-atom biological systems scale well up to ∼30k cores, producing ∼30 ns/day. Atomistic simulations of very large systems for time scales approaching the microsecond would, therefore, appear now to be within reach.

Electrostatics, structure prediction, and the energy landscapes for protein folding and binding

While being long in range and therefore weakly specific, electrostatic interactions are able to modulate the stability and folding landscapes of some proteins. The relevance of electrostatic forces for steering the docking of proteins to each other is widely acknowledged, however, the role of electrostatics in establishing specifically funneled landscapes and their relevance for protein structure prediction are still not clear. By introducing Debye-Hückel potentials that mimic long-range electrostatic forces into the Associative memory, Water mediated, Structure, and Energy Model (AWSEM), a transferable protein model capable of predicting tertiary structures, we assess the effects of electrostatics on the landscapes of thirteen monomeric proteins and four dimers. For the monomers, we find that adding electrostatic interactions does not improve structure prediction. Simulations of ribosomal protein S6 show, however, that folding stability depends monotonically on electrostatic strength. The trend in predicted melting temperatures of the S6 variants agrees with experimental observations. Electrostatic effects can play a range of roles in binding. The binding of the protein complex KIX-pKID is largely assisted by electrostatic interactions, which provide direct charge-charge stabilization of the native state and contribute to the funneling of the binding landscape. In contrast, for several other proteins, including the DNA-binding protein FIS, electrostatics causes frustration in the DNA-binding region, which favors its binding with DNA but not with its protein partner. This study highlights the importance of long-range electrostatics in functional responses to problems where proteins interact with their charged partners, such as DNA, RNA, as well as membranes.

Boundary condition effects on the dynamic and electric properties of hydration layers

Water solvation has a central role in several biochemical processes ranging from protein folding to biomolecular recognition and enzyme catalysis. Because of its importance, the structure and dynamics of hydration layers around biological macromolecules have been the targets of a great number of experimental and computational studies. In the present contribution, we have investigated the effects of periodic boundary conditions (PBCs), as used in conjunction with molecular dynamics (MD) simulations, on the dynamic and electric properties of water layers. In particular, we have systematically performed MD simulations of neat water and biomolecules in aqueous solutions by imposing a different external dielectric constant, a generally overlooked parameter in PBC simulations. The effect of the system size has also been addressed. Overall, our results consistently indicate that the dipole moment properties of water layers, and specifically the dipole moment fluctuations and the reorientational correlation functions, can be sensitive to the choice of the external boundary conditions, whereas other molecular properties, such as the self-diffusion coefficient and the reorientational relaxation times, are not affected. We think that our investigation may help to assess appropriate simulation conditions for modeling the aqueous environment of relevant biochemical systems and processes.

Influence of surface polarity on water dynamics at the water/rutile TiO₂(110) interface

We report molecular dynamics (MD) simulations of the water/clean rutile TiO2 (110) interface using polarizable and non-surface polarity force field models. The effect of surface polarity on the water dynamics near the TiO2(110) surface is addressed, specifically by calculating the water hydrogen bond and reorientational dynamics. The hydrogen bond lifetime of interfacial water molecules is several times longer than that of bulk water due to the strong water-TiO2 interactions. A comparison of the dynamics simulated with the polarizable and non-surface polarity models shows that, while the hydrogen bond lifetime between the interfacial water and TiO2 surface is insensitive to the surface polarity, the reorientational dynamics around this hydrogen bond axis is significantly influenced by the surface polarity; the surface polarity of the TiO2 increases the water-TiO2 interactions, stabilizing the local structure of the interfacial water molecules and restricting their rotational motion. This reorientation occurs predominantly by rotation around the O-H group hydrogen bonded to the TiO2 surface. Furthermore, we correlate the dynamics of the induced charge on the TiO2 surface with the interfacial water dynamics. Our results show that the timescale of correlations of the atom charges induced by the local electric field in bulk water is influenced by the rotational motion, hydrogen bond rearrangement and translational motion, while the induced charge dynamics of the TiO2 surface is governed primarily by the rotational dynamics of the interfacial water molecules. This study demonstrates that the solid surface polarity has a significant impact on the dynamics of water molecules near TiO2 surfaces.

An Overview of Electrostatic Free Energy Computations for Solutions and Proteins

Free energy simulations for electrostatic and charging processes in complex molecular systems encounter specific difficulties owing to the long-range, 1/r Coulomb interaction. To calculate the solvation free energy of a simple ion, it is essential to take into account the polarization of nearby solvent but also the electrostatic potential drop across the liquid-gas boundary, however distant. The latter does not exist in a simulation model based on periodic boundary conditions because there is no physical boundary to the system. An important consequence is that the reference value of the electrostatic potential is not an ion in a vacuum. Also, in an infinite system, the electrostatic potential felt by a perturbing charge is conditionally convergent and dependent on the choice of computational conventions. Furthermore, with Ewald lattice summation and tinfoil conducting boundary conditions, the charges experience a spurious shift in the potential that depends on the details of the simulation system such as the volume fraction occupied by the solvent. All these issues can be handled with established computational protocols, as reviewed here and illustrated for several small ions and three solvated proteins.

Design of a reaction field using a linear-combination-based isotropic periodic sum method

In our previous study (Takahashi et al., J. Chem. Theory Comput. 2012, 8, 4503), we developed the linear-combination-based isotropic periodic sum (LIPS) method. The LIPS method is based on the extended isotropic periodic sum theory that produces a ubiquitous interaction potential function to estimate homogeneous and heterogeneous systems. The LIPS theory also provides the procedure to design a periodic reaction field. To demonstrate this, in the present work, a novel reaction field of the LIPS method was developed. The novel reaction field was labeled LIPS-SW, because it provides an interaction potential function with a shape that resembles that of the switch function method. To evaluate the ability of the LIPS-SW method to describe in homogeneous and heterogeneous systems, we carried out molecular dynamics (MD) simulations of bulk water and water-vapor interfacial systems using the LIPS-SW method. The results of these simulations show that the LIPS-SW method gives higher accuracy than the conventional interaction potential function of the LIPS method. The accuracy of simulating water-vapor interfacial systems was greatly improved, while that of bulk water systems was maintained using the LIPS-SW method. We conclude that the LIPS-SW method shows great potential for high-accuracy, high-performance computing to allow large scale MD simulations. © 2014 Wiley Periodicals, Inc.

Dependence of protein crystal stability on residue charge states and ion content of crystal solvent

Protein crystallization is frequently induced by the addition of various precipitants, which directly affect protein solubility. In addition to organic cosolvents and long-chain polymers, salts belong to the most widely used precipitants in protein crystallography. However, despite such widespread usage, their mode of action at the atomistic level is still largely unknown. Here, we perform extensive molecular dynamics simulations of the villin headpiece crystal unit cell to examine its stability at different concentrations of sodium sulfate. We show that the inclusion of ions in crystal solvent at high concentration can prevent large rearrangements of the asymmetric units and a loss of symmetry of the unit cell without significantly affecting protein dynamics. Of importance, a similar result can be achieved by neutralizing several specific charged residues suggesting that they may play an active role in crystal destabilization due to unfavorable electrostatic interactions. Our results provide a microscopic picture behind salt-induced stabilization of a protein crystal and further suggest that adequate modeling of realistic crystallization conditions may be necessary for successful molecular dynamics simulations of protein crystals.

Isotropic Periodic Sum Treatment of Long-Range Electrostatic Interactions in Combined Quantum Mechanical and Molecular Mechanical Calculations

The isotropic periodic sum (IPS) method was extended to describe long-range electrostatic interactions in combined quantum mechanical and molecular mechanical (QM/MM) calculations. The resulting method, designated QM/MM-IPS, was tested for two ion association processes and a model SN2 reaction in aqueous solution. Potential of mean force (PMF) profiles and radial distribution functions computed from the QM/MM-IPS simulations were compared with those obtained by using the existing QM/MM-Ewald sum and cutoff (QM/MM-Cutoff) methods. In contrast to the QM/MM-Cutoff method, with which PMFs of ion separation tend to display a spurious linear drift, the QM/MM-IPS method successfully eliminates such artifacts, in excellent agreement with the QM/MM-Ewald results. The PMF obtained with the QM/MM-IPS method for the SN2 reaction that transfers an NH3 group between two chloride anions closely resembles that from the QM/MM-Ewald simulations. Compared with QM/MM-Ewald, the QM/MM-IPS method reduces the computational cost by 60-70% when a local region of 12 to 14 Å is used. These results suggest that the QM/MM-IPS method can be used as a reliable and efficient alternative to the QM/MM-Ewald method to incorporate long-range electrostatic effects in simulating solution-phase chemical reactions.

Quantifying Artifacts in Ewald Simulations of Inhomogeneous Systems with a Net Charge

Ewald summation, which has become the de facto standard for computing electrostatic interactions in biomolecular simulations, formally requires that the simulation box is neutral. For non-neutral systems, the Ewald algorithm implicitly introduces a uniform background charge distribution that effectively neutralizes the simulation box. Because a uniform distribution of counter charges typically deviates from the spatial distribution of counterions in real systems, artifacts may arise, in particular in systems with an inhomogeneous dielectric constant. Here, we derive an analytical expression for the effect of using an implicit background charge instead of explicit counterions, on the chemical potential of ions in heterogeneous systems, which (i) provides a quantitative criterium for deciding if the background charge offers an acceptable trade-off between artifacts arising from sampling problems and artifacts arising from the homogeneous background charge distribution, and (ii) can be used to correct this artifact in certain cases. Our model quantifies the artifact in terms of the difference in charge density between the non-neutral system with a uniform neutralizing background charge and the real neutral system with a physically correct distribution of explicit counterions. We show that for inhomogeneous systems, such as proteins and membranes in water, the artifact manifests itself by an overstabilization of ions inside the lower dielectric by tens to even hundreds kilojoules per mole. We have tested the accuracy of our model in molecular dynamics simulations and found that the error in the calculated free energy for moving a test charge from water into hexadecane, at different net charges of the system and different simulation box sizes, is correctly predicted by the model. The calculations further confirm that the incorrect distribution of counter charges in the simulation box is solely responsible for the errors in the PMFs.

Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: an accurate correction scheme for electrostatic finite-size effects

The calculation of a protein-ligand binding free energy based on molecular dynamics (MD) simulations generally relies on a thermodynamic cycle in which the ligand is alchemically inserted into the system, both in the solvated protein and free in solution. The corresponding ligand-insertion free energies are typically calculated in nanoscale computational boxes simulated under periodic boundary conditions and considering electrostatic interactions defined by a periodic lattice-sum. This is distinct from the ideal bulk situation of a system of macroscopic size simulated under non-periodic boundary conditions with Coulombic electrostatic interactions. This discrepancy results in finite-size effects, which affect primarily the charging component of the insertion free energy, are dependent on the box size, and can be large when the ligand bears a net charge, especially if the protein is charged as well. This article investigates finite-size effects on calculated charging free energies using as a test case the binding of the ligand 2-amino-5-methylthiazole (net charge +1 e) to a mutant form of yeast cytochrome c peroxidase in water. Considering different charge isoforms of the protein (net charges -5, 0, +3, or +9 e), either in the absence or the presence of neutralizing counter-ions, and sizes of the cubic computational box (edges ranging from 7.42 to 11.02 nm), the potentially large magnitude of finite-size effects on the raw charging free energies (up to 17.1 kJ mol(-1)) is demonstrated. Two correction schemes are then proposed to eliminate these effects, a numerical and an analytical one. Both schemes are based on a continuum-electrostatics analysis and require performing Poisson-Boltzmann (PB) calculations on the protein-ligand system. While the numerical scheme requires PB calculations under both non-periodic and periodic boundary conditions, the latter at the box size considered in the MD simulations, the analytical scheme only requires three non-periodic PB calculations for a given system, its dependence on the box size being analytical. The latter scheme also provides insight into the physical origin of the finite-size effects. These two schemes also encompass a correction for discrete solvent effects that persists even in the limit of infinite box sizes. Application of either scheme essentially eliminates the size dependence of the corrected charging free energies (maximal deviation of 1.5 kJ mol(-1)). Because it is simple to apply, the analytical correction scheme offers a general solution to the problem of finite-size effects in free-energy calculations involving charged solutes, as encountered in calculations concerning, e.g., protein-ligand binding, biomolecular association, residue mutation, pKa and redox potential estimation, substrate transformation, solvation, and solvent-solvent partitioning.

Charge-leveling and proper treatment of long-range electrostatics in all-atom molecular dynamics at constant pH

Recent development of constant pH molecular dynamics (CpHMD) methods has offered promise for adding pH-stat in molecular dynamics simulations. However, until now the working pH molecular dynamics (pHMD) implementations are dependent in part or whole on implicit-solvent models. Here we show that proper treatment of long-range electrostatics and maintaining charge neutrality of the system are critical for extending the continuous pHMD framework to the all-atom representation. The former is achieved here by adding forces to titration coordinates due to long-range electrostatics based on the generalized reaction field method, while the latter is made possible by a charge-leveling technique that couples proton titration with simultaneous ionization or neutralization of a co-ion in solution. We test the new method using the pH-replica-exchange CpHMD simulations of a series of aliphatic dicarboxylic acids with varying carbon chain length. The average absolute deviation from the experimental pK(a) values is merely 0.18 units. The results show that accounting for the forces due to extended electrostatics removes the large random noise in propagating titration coordinates, while maintaining charge neutrality of the system improves the accuracy in the calculated electrostatic interaction between ionizable sites. Thus, we believe that the way is paved for realizing pH-controlled all-atom molecular dynamics in the near future.

Behavior of solvent-exposed hydrophobic groove in the anti-apoptotic Bcl-XL protein: clues for its ability to bind diverse BH3 ligands from MD simulations

Bcl-XL is a member of Bcl-2 family of proteins involved in the regulation of intrinsic pathway of apoptosis. Its overexpression in many human cancers makes it an important target for anti-cancer drugs. Bcl-XL interacts with the BH3 domain of several pro-apoptotic Bcl-2 partners. This helical bundle protein has a pronounced hydrophobic groove which acts as a binding region for the BH3 domains. Eight independent molecular dynamics simulations of the apo/holo forms of Bcl-XL were carried out to investigate the behavior of solvent-exposed hydrophobic groove. The simulations used either a twin-range cut-off or particle mesh Ewald (PME) scheme to treat long-range interactions. Destabilization of the BH3 domain-containing helix H2 was observed in all four twin-range cut-off simulations. Most of the other major helices remained stable. The unwinding of H2 can be related to the ability of Bcl-XL to bind diverse BH3 ligands. The loss of helical character can also be linked to the formation of homo- or hetero-dimers in Bcl-2 proteins. Several experimental studies have suggested that exposure of BH3 domain is a crucial event before they form dimers. Thus unwinding of H2 seems to be functionally very important. The four PME simulations, however, revealed a stable helix H2. It is possible that the H2 unfolding might occur in PME simulations at longer time scales. Hydrophobic residues in the hydrophobic groove are involved in stable interactions among themselves. The solvent accessible surface areas of bulky hydrophobic residues in the groove are significantly buried by the loop LB connecting the helix H2 and subsequent helix. These observations help to understand how the hydrophobic patch in Bcl-XL remains stable in the solvent-exposed state. We suggest that both the destabilization of helix H2 and the conformational heterogeneity of loop LB are important factors for binding of diverse ligands in the hydrophobic groove of Bcl-XL.

Non-Ewald methods: theory and applications to molecular systems

Several non-Ewald methods for calculating electrostatic interactions have recently been developed, such as the Wolf method, the reaction field method, the pre-averaging method, and the zero-dipole summation method, for molecular dynamics simulations of various physical systems, including biomolecular systems. We review the theories of these approaches and their potential applications to molecular simulations, and discuss their relationships.